"Parrot" works in American English as well, if you are talking about their speech. If you are talking about their character, "sheep" is good. We have many terms, depending on the topic (politics, culture, religion) and the speaker's point of view who is labeling the other person. "Parrot" and "sheep" are good general terms.

The British was talking about homosexuality and that he was a sheep repeating what other people say but now he is not against gays though he is not gay himself.
Here he pointed to his speech not character but yet used the word sheep?

I would say that he actually referred to his character: sheep stick with their flock and don't stray. Likewise, this man did what the others did and voiced a certain opinion about gays without actually stopping to think for himself. Thus, he was a sheep.

I don't think we could call the person himself a "parrot" but we might use "parrot" as a verb to indicate someone was just repeating something - e.g. "He's just parroting receive opinion." You can find quite a few examples by Googling. Here's one:
"The reporter's job is to do research to find the facts. But too often they seem to parrot back whatever is fed to them by press releases, politicians, or other news reports"

The person is more like a "sheep." So I guess we could say that "sheep are people who just parrot others' ideas." There's a rather mixed metaphor!